Understanding the Basics of Extended Plane and Space

[*best&sweetest*]

Could anybody please explain the concept of extended plane and extended space? I know the definition of an ideal point and that's the only part that I really understand. If each two distinct planes (in extended space) meet in just one line, does it mean that there's only one ideal line for each plane?
What's the difference between extended plane and extended space? There's only one extended plane in extended space, right?
I know that for some of you these questions will sound very silly, but the book I have has a very poor explanation. If you know some web page where I can learn the basics, please tell me. Thank you!

 

[HallsofIvy]

I don't believe the term "extended space" is standard. There are many ways to "extend space".

If each two distinct planes (in extended space) meet in just one line, does it mean that there's only one ideal line for each plane?

Yes, that's true. In particular planes that are "parallel" in regular space meet in that one "ideal" line (and a pair of lines, one on each plane, meet in an "ideal" point on that "ideal" line.

What's the difference between extended plane and extended space? There's only one extended plane in extended space, right?

No, every plane in regular space corresponds to a different extended plane in extended space.

 

[tacman]

Extended plane and extended space are concepts used in geometry to expand the traditional Euclidean space. In traditional Euclidean geometry, a plane is a flat surface that extends infinitely in all directions and has no thickness. Similarly, space is a three-dimensional extension of this concept, with length, width, and height.

In extended plane and extended space, these concepts are extended further to include ideal points and lines. Ideal points are points that do not have a physical location but are used to represent points at infinity. They are often denoted by the symbol "∞" and are used to extend lines and planes infinitely.

In extended space, each two distinct planes will meet at a single line. This means that there is only one ideal line for each plane. This is because in extended space, all points at infinity are considered to be the same point. So, any line that extends to infinity from a plane will intersect with all other lines that extend from other planes at the same point at infinity.

The main difference between extended plane and extended space is that extended plane only includes ideal points and lines in a two-dimensional plane, while extended space includes these concepts in a three-dimensional space. In extended space, there is only one extended plane, as all planes are considered to be parallel to each other and intersect at infinity.

It is important to note that extended plane and extended space are theoretical concepts and are not physically tangible. They are used in geometry to expand the traditional concepts of points, lines, and planes and to aid in solving complex geometric problems.

If you are looking for more information on extended plane and extended space, there are many online resources available. You can start by searching for "extended plane and extended space" or "ideal points and lines in geometry" to find articles, videos, and other materials that can help you understand these concepts in more depth. Additionally, consulting a geometry textbook or seeking guidance from a math teacher or tutor can also be helpful in understanding these concepts.